| Volume 9, Number 4, 2019, Pages 1333-1346 DOI：10.11948/2156-907X.20180238 |
| Bogdanov-Takens bifurcation in a delayed Michaelis-Menten type ratio-dependent predator-prey system with prey harvesting |
| Yunxian Dai,Ping Yang,Zhiliang Luo,Yiping Lin |
| Keywords:Delayed ratio-dependent predator-prey model, Michaelis-Menten type, prey harvesting, Bogdanov-Takens bifurcation. |
| Abstract: |
| In this paper, we study a delayed Michaelis-Menten Type ratio-dependent predator-prey model with prey harvesting. By considering the characteristic equation associated with the nonhyperbolic equilibrium, the critical value of the parameters for the Bogdanov-Takens bifurcation is obtained. The conditions for the characteristic equation having negative real parts are discussed. Using the normal form theory of Bogdanov-Takens bifurcation for retarded functional differential equations, the corresponding normal form restricted to the associated two-dimensional center manifold is calculated and the versal unfolding is considered. The parameter conditions for saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained. Numerical simulations are given to support the analytical results. |
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